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<p>This paper focuses on high-order numerical methods for backward stochastic differential equations and proposes a novel one-step high-order scheme based on multi-stage predictor-corrector techniques. The proposed one-step high-order scheme integrates an initial prediction step with implicit multi-stage correction steps, offering the advantage of flexibly achieving adjustable high convergence orders through the number of stages while completing all necessary calculations within a single time step. Rigorous stability analysis of the proposed scheme is conducted to demonstrate that numerical errors remain bounded and do not grow exponentially with increasing time steps. Furthermore, strict error estimates are established via Itô-Taylor expansions to confirm the one-step high-order convergence property. Finally, numerical experiments on both linear and nonlinear backward stochastic differential equations validate the theoretical findings, showing good agreement between the theoretical convergence rates and practical results. Existing methods such as Euler schemes, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\theta\)</EquationSource> </InlineEquation>-schemes, and other multi-stage schemes are shown to be special cases of the proposed framework.</p>

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A novel one-step high-order scheme for backward stochastic differential equations based on the multi-stage and predictor-corrector technology

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Qiang Han,
  • Qianwen Hu,
  • Quanxin Zhu

摘要

This paper focuses on high-order numerical methods for backward stochastic differential equations and proposes a novel one-step high-order scheme based on multi-stage predictor-corrector techniques. The proposed one-step high-order scheme integrates an initial prediction step with implicit multi-stage correction steps, offering the advantage of flexibly achieving adjustable high convergence orders through the number of stages while completing all necessary calculations within a single time step. Rigorous stability analysis of the proposed scheme is conducted to demonstrate that numerical errors remain bounded and do not grow exponentially with increasing time steps. Furthermore, strict error estimates are established via Itô-Taylor expansions to confirm the one-step high-order convergence property. Finally, numerical experiments on both linear and nonlinear backward stochastic differential equations validate the theoretical findings, showing good agreement between the theoretical convergence rates and practical results. Existing methods such as Euler schemes, \(\theta\) -schemes, and other multi-stage schemes are shown to be special cases of the proposed framework.